56 problems

Let $\xi$ be an irrational real number whose base-$b$ expansion is generated by a finite automaton; equivalently, its digit sequence is $k$-automatic for some integer base…

Let $q$ be a positive power of a prime, and let $\textbf{S}$ be an automatic sequence over $\mathbb{F}_q$. Let $W_q(\textbf{S})$ denote the number wall associated with $\textbf{S}$…

Let $cmathbf a=(a_1,\cdots,a_k)$ be a positive integral vector of dimension $k>1$, and let $f(cmathbf x)=cmathbf acdotcmathbf x$ be its linear form. Let $A_n$ be the anti-recurrence…

An anti-recurrence sequence is the increasing sequence generated by a positive integral linear form, by selecting the integers represented by that form without repetition. A linear…

For an irrational number $\xi$, define the deterministic random walk by … Call $r$ a record if $S_r(2\xi)$ has not occurred among the preceding partial sums, including…

For $g\geq 2$, let $S_{1,g,g+1}$ be the greedy 3-subsumfree sequence beginning with $1,g,g+1$, where each subsequent term is the smallest larger integer that is not the sum of thre…

Let $S_{1,2,3}$ be the greedy 3-subsumfree sequence beginning with $1,2,3$: after the initial values, each term is the smallest larger integer that is not the sum of three distinct…

Let $X_n$ be the sequence of anti-$k$-naccis, defined as sums of $k$ consecutive missing numbers. A sequence is $k$-automatic when its base-$k$ value sequence can be generated by a…

The Fibonacci-index formula conjecture.

The conjectural formulas for earliest Thue–Morse progressions.

Cloitre's density conjecture. The number of $1$'s appearing in the prefix $a_1a_2\cdots a_n$ is $2n/3+o(n)$.

Let $t\geq 1$ be an odd integer, and let $c_t$ denote the correlation quantity associated with the binary sum-of-digits difference for $t$. For each $K\geq 1$, consider the integer…

Cusick's conjecture. For all $t\geq 0$ we have

Additive complexity conjecture. The additive complexity of a $k$-automatic sequence is a $k$-regular sequence.

Let $u$ be a binary word of length $n$ of the form $0^a1^{2^k}0^b$, where $a,b,k\in\mathbb{Z}_{\geq1}$. Power-of-two block conjecture. The word $u$ belongs to $\mathscr{U}$ if and…

Let $k$ and $q$ be positive integers, and let a sequence over $\mathcal{F}_q$ be $k$-automatic. Its number wall is the associated two-dimensional array of Hankel determinants. Auto…

Let $\mathcal{\Upsilon}(t)$ be the Laurent series associated with the Paperfolding sequence, and let $t$-$LC$ denote the $p(t)$-adic Littlewood Conjecture with $p(t)=t$. Adiceam–Nes…

Let $K$ be a global function field with normalized valuation $v$, let $K_v$ be its completion, and let $\pi\in K_v$ satisfy $v(\pi)>0$. Let $A_v$ be the completion of the valuation…

The fixed-point basis conjecture. For every $i\in\{1,2,\dots,k\}$, the polynomial

The dimensions conjecture. For every divisor $m$ of $\operatorname{lcm}(\deg R_1,\dots,\deg R_k)$, this vector space has dimension

Omega correlation conjecture.

Let $k,\ell\in\mathbb{N}$ be multiplicatively independent, meaning that no positive powers of $k$ and $\ell$ are equal. Let $\mathbf a=(a_n)_{n\in\mathbb{N}_0}$ be an integer seque…

Let $V$ be a vector subspace of $\bigoplus_{i=1}^\infty \mathbb{Q}$. For a prime $p$, let $\nu_p$ denote the $p$-adic valuation, and let $p_1,p_2,\dots$ be the increasing enumerati…

Let $c$ be a finite word satisfying condition (WH), and assume that one of the equivalent assertions in Proposition 4 holds. Let $\mathbf{u}$ be the associated simple-Parry automat…

Let $x_{3/2}$ be the binary Thue–Morse sequence in base $3/2$. For a finite word $w=w_1\dots w_m$, write $\overleftarrow{w}=w_m\dots w_1$ for its reversal. Reversal invariance conj…